The orthogonal polynomials method using Gegenbauer polynomials to solve mixed integral equations with a Carleman kernel

The orthogonal polynomials approach with Gegenbauer polynomials is an effective tool for analyzing mixed integral equations (MIEs) due to their orthogonality qualities. This article reviewed recent breakthroughs in the use of Gegenbauer polynomials to solve mixed integral problems. Previous authors studied the problem with a continuous kernel that combined both Volterra (V) and Fredholm (F) components; however, in this paper, we focused on a singular Carleman kernel. The kernel of FI was measured with respect to position in the space $L_2[-1,1],$ while the kernel of Ⅵ was considered as a function of time in the space $C[0,T],T<1$. The existence of a unique solution was discussed in $L_2\left[-1,1\right]\times C\left[0,T\right]$ space. The solution and its error stability were both investigated and commented on. Finally, numerical examples were reviewed, and their estimated errors were assessed using Maple (2022) software.